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Exam 6: The Trigonometric Functions

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Exam 1: Fundamental Concepts of Algebra150 Questionsadd course
Exam 2: Equations and Inequalities142 Questionsadd course
Exam 3: Functions and Graphs147 Questionsadd course
Exam 4: Polynomial and Rational Functions147 Questionsadd course
Exam 5: Inverse, Exponential, and Logarithmic Functions144 Questionsadd course
Exam 6: The Trigonometric Functions150 Questionsadd course
Exam 7: Analytic Trigonometry150 Questionsadd course
Exam 8: Applications of Trigonometry144 Questionsadd course
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The phases of the moon can be described using the phase angle The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:

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The phases of the moon can be described using the phase angle The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:

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Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points.

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Approximate, to the nearest 0.01 radian, all angles Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.0135 in the interval [ 0, 2 Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.0135 ) that satisfy the equation. Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.0135 = 0.0135

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Question 4

Scientists sometimes use the formula Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later. to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later. , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later. occurs at 2 P.M., and the average temperature of 21 Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later. occurs 6 hours later.

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Question 5

Approximate to four decimal places. Approximate to four decimal places.

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Question 6

Verify the identity by transforming the left-hand side into the right-hand side. Verify the identity by transforming the left-hand side into the right-hand side.

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Question 7

Find y by referring to the graph of the trigonometric function. Find y by referring to the graph of the trigonometric function.

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Question 8

Verify the identity by transforming the left-hand side into the right-hand side. Verify the identity by transforming the left-hand side into the right-hand side.

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Question 9

Find the exact values of the six trigonometric functions of the angle, whenever possible. Find the exact values of the six trigonometric functions of the angle, whenever possible.

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Question 10

Find the quadrant containing Find the quadrant containing if the given conditions are true. and if the given conditions are true. Find the quadrant containing if the given conditions are true. and and Find the quadrant containing if the given conditions are true. and

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Question 11

An airplane flying at a speed of 300 mi/hr flies from a point A in the direction An airplane flying at a speed of 300 mi/hr flies from a point A in the direction for 15 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A. for 15 minutes and then flies in the direction An airplane flying at a speed of 300 mi/hr flies from a point A in the direction for 15 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A. for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.

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Question 12

Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. in degrees and D in meters is given by Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. , where Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. meter, its resolution is Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. . Approximate the diameter of the lens to the nearest hundredth of a meter. Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.

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Question 13

Scientists sometimes use the formula Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 , and the low temperature of -25 occurs at 4 A.M. to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 , and the low temperature of -25 occurs at 4 A.M. , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 , and the low temperature of -25 occurs at 4 A.M. , and the low temperature of -25 Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 , and the low temperature of -25 occurs at 4 A.M. occurs at 4 A.M.

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Question 14

Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , . , find Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , . , Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , . , Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , . , Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , . .

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Question 15

A rocket is fired at sea level and climbs at a constant angle of A rocket is fired at sea level and climbs at a constant angle of through a distance of 10,500 feet. Approximate its altitude to the nearest foot. through a distance of 10,500 feet. Approximate its altitude to the nearest foot.

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Question 16

The graph of an equation of a sine wave is shown in the figure. Find the amplitude, period, and phase shift. The graph of an equation of a sine wave is shown in the figure. Find the amplitude, period, and phase shift.

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Question 17

An airplane flying at a speed of 360 mi/hr flies from a point A in the direction An airplane flying at a speed of 360 mi/hr flies from a point A in the direction for 45 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A. for 45 minutes and then flies in the direction An airplane flying at a speed of 360 mi/hr flies from a point A in the direction for 45 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A. for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.

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Question 18

Find the period of the equation. Find the period of the equation.

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Question 19

Express the angle as a decimal, to the nearest ten-thousandth of a degree. Express the angle as a decimal, to the nearest ten-thousandth of a degree.

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Question 20
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